Hiking Again

In about an hour and a half I will go hiking again, this time in the mountains towards a waterfall. Sadly, we cannot play in the waterfall since people have died there before. I’m looking forward to the hike.

I thought I might quickly summarize some of the things I have learned here in Utah for my own health. I will create a boring list below:

I have learned why hyperbolic geometry implies a low energy density of the universe relative to Euclidean geometry, which in turn provides a low energy density relative to spherical geometry. It has to do with using large textbooks to smash things like eggshells and pringles. Long story.

I have learned a lot about what is called Group Action, which is the idea of using elements of a group to act on elements of another set. I can define the action of a group G on a set X by the rule g*x=x’, where x and x’ are elements of x. So we can think of each element of G as a function from X to X. I can define this action so long as the identity element 1 of the group acts as the identity function from X to X and (gh)*x=g*(h*x). (in other words, acting by the combination of elements of G is the same as composing). The stabilizer of an element x in X is the set of elements of G that send x to itself. In other words, Stab(x)={g in G : g*x=x}. the orbit of an element x in X is the set of possible images of x under elements of G. Formally, Orb(x)={x’ in X : g*x=x’ for some g in G}. The orbit of x is the answer to the question, “where can elements of G take this x?” If elements of G can take a given x in X to any x’ in X, then we say that G acts transitively on X. If any x in X can get to an element x’ in its orbit by way of only one unique element of g, then we say G acts simply on X. If G acts simply and transitively, then it acts simply transitively. So yeah, actions are really neat to think about. We have been doing a special case of actions, where the elements of our group are matrices and the actions are defined by matrix multiplication (or sometimes something vaguely related to matrix multiplication). That’s because a class of Lie groups called Linear Lie Groups are precisely groups of matrices with non-zero determinant.

Another big idea that I’ve learned is that topological spaces can be groups too. We’ve been talking about Lie groups, which can be thought of as smooth manifolds with a group structure. This is sort of a novel idea to me. Having learned these concepts separately, I never really thought about combining them together except briefly when I thought about how a topology is a monoid under unions. The geometry of it is also really interesting. We take a quotient group, for instance R mod Z under addition, and we get the interval [0,1] with 0 being identified with 1. So, we glue together 0 and 1 to get a circle. The circle can be given a topology from the interval if we would like. The interval in this case is called the fundamental domain. The same trick can be done with R^2 mod Z^2 to get a square instead of the interval, except with the bottom of the square identified with the top, and the left with the right. If you glue the left and right sides of a square, you get a cylinder, and then if you glue the top and bottom of the cylinder together you get a donut-shaped object we call a torus. I learned these things in topology, but I never realized that they could arise from the quotients of groups. Things get more interesting when we connect topology, groups, and geometry. We can get geometric spaces from groups too. We can get a many-holed torus to fit together only if we put it in hyperbolic space, since the angles in a regular polygon can get as small as we like. We then get a many holed torus with a hyperbolic metric and we can give it a topology as well from the gluing process. I don’t yet understand all of the nuance that happens in the interplay between topology, metric spaces, geometry, and groups, but I’ve had a taste and it tastes good.

The last idea I’ll write about is that of smooth manifolds. A smooth manifold is a manifold with a smooth structure to it. Go figure. A manifold is a topological space that is Hausdorff (meaning our points aren’t squished together too tightly), and also looks locally like Euclidean space. That means every point in our manifold has a tiny neighborhood around it that looks like a small disk in R^n . The smoothness comes from the fact that any two neighborhoods around a point that each look like R^n have a piece that is diffeomorphic to the other piece. This means that there is an invertible function between them that one can take as many derivatives of as one wishes and whose inverse also has infinitely many derivatives. We call functions smooth when all the derivatives exist. This brings up another important idea I learned. I learned what a full derivative in higher dimensions is. It’s a matrix of partial derivatives that sends the space of vectors around a point in the domain to the tangent space of the image of that point. In one dimension it gives the slope of the tangent line, and in three dimensions using two variables, it gives the tangent plane touching the surface at a given point. Very cool indeed.

There is some more to write about, namely that one can put even more structure on a smooth manifold to get geometry (curvature). This additional structure is an inner product, which let’s one define distance, curvature, and abstract calculus. However, I have to go on a hike. Bis später!

Oh I almost forgot.I learned about the Cantor set and Cantor function from my room mate Jesse. Blows my mind. An uncountable set of measure zero and a uniformly continuous function on the compliment of the set.

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